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A ternary Diophantine inequality with prime numbers of a special form

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Abstract

Let N be a sufficiently large real number. In this paper, we prove that, for \(1<c<\frac{973}{856}\) and for any arbitrarily large number \(E>0\), the Diophantine inequality

$$\begin{aligned} \big |p_1^c+p_2^c+p_3^c-N\big |<(\log N)^{-E} \end{aligned}$$

is solvable in prime variables \(p_1,p_2,p_3\) such that each of the numbers \(p_i+2\) for \(i=1,2,3\) has at most \([\frac{12626}{4865-4280c}]\) prime factors counted with multiplicity. This result constitutes an improvement upon the previous result of Zhu (Proc Indian Acad Sci Math Sci 130(1):23, 2020).

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Acknowledgements

The authors would like to express their most sincere gratitude to Professor Wenguang Zhai for his valuable advice and constant encouragement. Also, the authors appreciate the referee for his/her patience in refereeing this paper. This work is supported by the National Natural Science Foundation of China (Grants No. 11901566, 12001047, 11971476, 12071238), the Fundamental Research Funds for the Central Universities (Grant No. 2021YQLX02), the National Training Program of Innovation and Entrepreneurship for Undergraduates (Grant No. 202107010), the Undergraduate Ed- ucation and Teaching Reform and Research Project for China University of Mining and Technology (Beijing) (Grant No. J210703), and the Scientic Research Funds of Beijing Information Science and Technology University (Grant No. 2025035).

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Jinjiang Li & Fei Xue

  2. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, People’s Republic of China

    Min Zhang

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  1. Jinjiang Li
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  2. Fei Xue
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  3. Min Zhang
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Correspondence to Min Zhang.

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Li, J., Xue, F. & Zhang, M. A ternary Diophantine inequality with prime numbers of a special form. Period Math Hung 85, 14–31 (2022). https://doi.org/10.1007/s10998-021-00415-9

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